Time-dependent Backward Stochastic Evolution Equations

We consider the following infinite dimensional backward stochastic evolution equation:{−dY (t) = (A(t) Y (t) + f(t, Y (t),Z(t) dt − Z(t) dW(t),Y (T) =(Xi),where A(t), t ≥ 0, are unbounded operators that generate a strong evolution operator U(t, r), 0 ≤ r ≤ t ≤ T. We prove under non-Lipschitz conditions that such an equation admits a unique evolution solution. Some examples and regularity properties of this solution are given as well